Bridge Hopping on Conducting Polymers in Solution

Configurational fluctuations of conducting polymers in solution can bring into proximity monomers which are distant from each other along the backbone. Electrons can hop between these monomers across the “bridges” so formed. We show how this can lead to (i) a collapse transition for metallic polymers, and (ii) to the observed dramatic efficiency of acceptor molecules for quenching fluorescence in semiconducting polymers. 71.20.Hk, 61.41.+e Typeset using REVTEX 1 Conducting (conjugated) polymers have been of great interest because of their unusual electrical and optical properties, combined with mechanical features very different from those of metallic conductors [1]. In solution, where the polymers are flexible, the strong conformational fluctuations can modify both the structural and electronic properties [2,3,5,4,6,7], which in suitable cases may then be sensitively controlled by environmental parameters, such as the temperature. Theoretical studies have been limited largely to numerical methods. But in an earlier paper [7] we have demonstrated the applicability of functional integral methods to these systems, allowing analytic treatment up to very late stages of the calculation, with the usual attendant advantages, including the possibility of treating realistically long chains, which are inaccessible in practice to the numerical methods. Here we concentrate on the effects of “bridge conduction”, the hopping of electrons between monomers distant from one another as measured along the polymer backbone, but close to each other in configuration space as a consequence of the bending of the dissolved polymer. This bridge conduction is believed to be one of the important mechanisms for electron transfer reactions which play a key role in several biological processes, such as photosynthesis and cell metabolism [9,10]. A similar phenomenon is expected for many chains in solution, between monomers on different chains which approach one another, particularly when they are stretched [11]. Bridge hopping was discussed briefly in a paper by Otto and Vilgis [13], but their results suffer from a critical neglect of the fermion statistics of the electrons. We focus here on two characteristic consequences of bridge hopping. The first, for metallic polymers (partially filled bands) is the resultant effective attraction between remote monomers, and the consequent contribution to the tendency for collapse from a swollen state. That collapse has major consequences for the electronic and optical properties of the polymer. The second, for semiconducting polymers, results from the dramatically enhanced rate for locally excited carriers to reach remote parts of the chain, by way of the bridges. We show that this reasonably explains the observed [8] ability of a single molecule to quench the fluorescence of a full chain of hundreds of monomers or more on time scales of picoseconds, 2 orders of magnitude shorter than the carrier diffusion time along the chain. Because our goal is limited to exploring these impacts of bridge hopping, we have chosen the simplest possible model. We neglect completely the dependence of the electronic hopping along the chain on the local configuration, which dependence was, in fact, the focus of our earlier paper. We also take no explicit account of chain rigidity associated with the moduli of twisting or bending. Rather, we make the usual rescaling of the monomer unit to a Kuhn, or persistence, length, with the new effective units executing a random self-avoiding walk. Consider a polymer chain on which a gas of electrons can hop. The polymer is supposed to be on a d-dimensional lattice with lattice parameter equal to the Kuhn length a. Then the partition function is the sum over self-avoiding walks for the polymer of the square of an electronic partition function, Zel({rn}) (to account for the spin of the electrons) , where rn describes the position of the n th lattice site visited Z = ∑ SAW Z el({rn}) (1) The electronic partition function can be written as a functional integral over Fermionic (Grassmanian) fields [12]: Zel({rn}) = ∫ cn(β)=−cn(0) D(cn (t), cn(t)) exp (